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The question is asking to find two two-digit numbers whose Highest Common Factor (H.C.F.) is 16 and Least Common Multiple (L.C.M.) is 480.

To solve this, we use the relationship between the H.C.F. (or GCD) and L.C.M. of two numbers \(a\) and \(b\):

\[
\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b
\]

In this case, we are given:

\[
\text{HCF} = 16, \quad \text{LCM} = 480
\]

Thus, the product of the two numbers will be:

\[
a \times b = 16 \times 480 = 7680
\]

Now, we need to find two numbers whose product is 7680 and whose H.C.F. is 16. Let’s express these numbers as:

\[
a = 16 \times x, \quad b = 16 \times y
\]

This implies that:

\[
a \times b = (16 \times x) \times (16 \times y) = 16^2 \times x \times y
\]

Given that \(a \times b = 7680\), we get:

\[
16^2 \times x \times y = 7680
\]

\[
x \times y = \frac{7680}{256} = 30
\]

Now we need to find two factors of 30 that are coprime (since the H.C.F. of \(x\) and \(y\) must be 1). The pair of factors that satisfy this condition are \(x = 5\) and \(y = 6\).

Thus, the numbers are:

\[
a = 16 \times 5 = 80, \quad b = 16 \times 6 = 96
\]

So, the two numbers are **80** and **96**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the relationship between H.C.F. and L.C.M. derived?
2. What is the process for finding the H.C.F. of two numbers using the Euclidean algorithm?
3. Can the method used here be applied to non-integer numbers?
4. How would you solve this problem if more than two numbers were involved?
5. What are the properties of numbers whose H.C.F. is 1?

**Tip:** Always check if the factors you're choosing are relatively prime when using this method for solving H.C.F. and L.C.M. problems.

The H.C.F. and L.C.M. of two two-digit numbers are 16 and 480, respectively. The numbers are?

Solve a problem involving two two-digit numbers where the HCF is 16 and the LCM is 480. Learn about the relationship between HCF and LCM, prime factorization, and how to find two numbers using this method. Suitable for students in Grades 8-10.

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